CS236781: Deep Learning on Computational Accelerators

Homework Assignment 2

Faculty of Computer Science, Technion.

Submitted by:

# Name Id email
Student 1 Yiftach Edelstain 316089952 yiftach.ede@campus.technion.ac.il
Student 2 Gur Telem 206631848 gurt@campus.technion.ac.il

Introduction

In this assignment we'll create a from-scratch implementation of two fundemental deep learning concepts: the backpropagation algorithm and stochastic gradient descent-based optimizers. Following that we will focus on convolutional networks with residual blocks. We'll use PyTorch to create our own network architectures and train them using GPUs on the course servers, and we'll conduct architecture experiments to determine the the effects of different architectural decisions on the performance of deep networks.

General Guidelines

Contents

$$ \newcommand{\mat}[1]{\boldsymbol {#1}} \newcommand{\mattr}[1]{\boldsymbol {#1}^\top} \newcommand{\matinv}[1]{\boldsymbol {#1}^{-1}} \newcommand{\vec}[1]{\boldsymbol {#1}} \newcommand{\vectr}[1]{\boldsymbol {#1}^\top} \newcommand{\rvar}[1]{\mathrm {#1}} \newcommand{\rvec}[1]{\boldsymbol{\mathrm{#1}}} \newcommand{\diag}{\mathop{\mathrm {diag}}} \newcommand{\set}[1]{\mathbb {#1}} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \newcommand{\pderiv}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\bb}[1]{\boldsymbol{#1}} $$

Part 1: Backpropagation

In this part we will learn about backpropagation and automatic differentiation. We'll implement both of these concepts from scratch and compare our implementation to PyTorch's built in implementation (autograd).

The backpropagation algorithm is at the core of training deep models. To state the problem we'll tackle in this notebook, imagine we have an L-layer MLP model, defined as $$ \hat{\vec{y}^i} = \vec{y}_L^i= \varphi_L \left( \mat{W}L \varphi{L-1} \left( \cdots \varphi_1 \left( \mat{W}_1 \vec{x}^i + \vec{b}_1 \right) \cdots \right)

a pointwise loss function $\ell(\vec{y}, \hat{\vec{y}})$ and an empirical loss over our entire data set, $$ L(\vec{\theta}) = \frac{1}{N} \sum_{i=1}^{N} \ell(\vec{y}^i, \hat{\vec{y}^i}) + R(\vec{\theta}) $$

where $\vec{\theta}$ is a vector containing all network parameters, e.g. $\vec{\theta} = \left[ \mat{W}_{1,:}, \vec{b}_1, \dots, \mat{W}_{L,:}, \vec{b}_L \right]$.

In order to train our model we would like to calculate the derivative (or gradient, in the multivariate case) of the loss with respect to each and every one of the parameters, i.e. $\pderiv{L}{\mat{W}_j}$ and $\pderiv{L}{\vec{b}_j}$ for all $j$. Since the gradient "points" to the direction of functional increase, the negative gradient is often used as a descent direction for descent-based optimization algorithms. In other words, iteratively updating each parameter proportianally to it's negetive gradient can lead to convergence to a local minimum of the loss function.

Calculus tells us that as long as we know the derivatives of all the functions "along the way" ($\varphi_i(\cdot),\ \ell(\cdot,\cdot),\ R(\cdot)$) we can use the chain rule to calculate the derivative of the loss with respect to any one of the parameter vectors. Note that if the loss $L(\vec{\theta})$ is scalar (which is usually the case), the gradient of a parameter will have the same shape as the parameter itself (matrix/vector/tensor of same dimensions).

For deep models that are a composition of many functions, calculating the gradient of each parameter by hand and implementing hard-coded gradient derivations quickly becomes infeasible. Additionally, such code makes models hard to change, since any change potentially requires re-derivation and re-implementation of the entire gradient function.

The backpropagation algorithm, which we saw in the lecture, provides us with a effective method of applying the chain rule recursively so that we can implement gradient calculations of arbitrarily deep or complex models.

We'll now implement backpropagation using a modular approach, which will allow us to chain many components layers together and get automatic gradient calculation of the output with respect to the input or any intermediate parameter.

To do this, we'll define a Layer class. Here's the API of this class:

In other words, a Layer can be anything: a layer, an activation function, a loss function or generally any computation that we know how to derive a gradient for.

Each block must define a forward() function and a backward() function.

Here's a diagram illustrating the above explanation:

Note that the diagram doesn't show that if the function is parametrized, i.e. $f(\vec{x},\vec{y})=f(\vec{x},\vec{y};\vec{w})$, there are also gradients to calculate for the parameters $\vec{w}$.

The forward pass is straightforward: just do the computation. To understand the backward pass, imagine that there's some "downstream" loss function $L(\vec{\theta})$ and magically somehow we are told the gradient of that loss with respect to the output $\vec{z}$ of our block, i.e. $\pderiv{L}{\vec{z}}$.

Now, since we know how to calculate the derivative of $f(\vec{x},\vec{y};\vec{w})$, it means we know how to calculate $\pderiv{\vec{z}}{\vec{x}}$, $\pderiv{\vec{z}}{\vec{y}}$ and $\pderiv{\vec{z}}{\vec{w}}$ . Thanks to the chain rule, this is all we need to calculate the gradients of the loss w.r.t. the input and parameters:

$$ \begin{align} \pderiv{L}{\vec{x}} &= \pderiv{L}{\vec{z}}\cdot \pderiv{\vec{z}}{\vec{x}}\\ \pderiv{L}{\vec{y}} &= \pderiv{L}{\vec{z}}\cdot \pderiv{\vec{z}}{\vec{y}}\\ \pderiv{L}{\vec{w}} &= \pderiv{L}{\vec{z}}\cdot \pderiv{\vec{z}}{\vec{w}} \end{align} $$

Comparison with PyTorch

PyTorch has the nn.Module base class, which may seem to be similar to our Layer since it also represents a computation element in a network. However PyTorch's nn.Modules don't compute the gradient directly, they only define the forward calculations. Instead, PyTorch has a more low-level API for defining a function and explicitly implementing it's forward() and backward(). See autograd.Function. When an operation is performed on a tensor, it creates a Function instance which performs the operation and stores any necessary information for calculating the gradient later on. Additionally, Functionss point to the other Function objects representing the operations performed earlier on the tensor. Thus, a graph (or DAG) of operations is created (this is not 100% exact, as the graph is actually composed of a different type of class which wraps the backward method, but it's accurate enough for our purposes).

A Tensor instance which was created by performing operations on one or more tensors with requires_grad=True, has a grad_fn property which is a Function instance representing the last operation performed to produce this tensor. This exposes the graph of Function instances, each with it's own backward() function. Therefore, in PyTorch the backward() function is called on the tensors, not the modules.

Our Layers are therefore a combination of the ideas in Module and Function and we'll implement them together, just to make things simpler. Our goal here is to create a "poor man's autograd": We'll use PyTorch tensors, but we'll calculate and store the gradients in our Layers (or return them). The gradients we'll calculate are of the entire block, not individual operations on tensors.

To test our implementation, we'll use PyTorch's autograd.

Note that of course this method of tracking gradients is much more limited than what PyTorch offers. However it allows us to implement the backpropagation algorithm very simply and really see how it works.

Let's set up some testing instrumentation:

Notes:

Layer Implementations

We'll now implement some Layers that will enable us to later build an MLP model of arbitrary depth, complete with automatic differentiation.

For each block, you'll first implement the forward() function. Then, you will calculate the derivative of the block by hand with respect to each of its input tensors and each of its parameter tensors (if any). Using your manually-calculated derivation, you can then implement the backward() function.

Notice that we have intermediate Jacobians that are potentially high dimensional tensors. For example in the expression $\pderiv{L}{\vec{w}} = \pderiv{L}{\vec{z}}\cdot \pderiv{\vec{z}}{\vec{w}}$, the term $\pderiv{\vec{z}}{\vec{w}}$ is a 4D Jacobian if both $\vec{z}$ and $\vec{w}$ are 2D matrices.

In order to implement the backpropagation algorithm efficiently, we need to implement every backward function without explicitly constructing this Jacobian. Instead, we're interested in directly calculating the vector-Jacobian product (VJP) $\pderiv{L}{\vec{z}}\cdot \pderiv{\vec{z}}{\vec{w}}$. In order to do this, you should try to figure out the gradient of the loss with respect to one element, e.g. $\pderiv{L}{\vec{w}_{1,1}}$ and extrapolate from there how to directly obtain the VJP.

Activation functions

(Leaky) ReLU

ReLU, or rectified linear unit is a very common activation function in deep learning architectures. In it's most standard form, as we'll implement here, it has no parameters.

We'll first implement the "leaky" version, defined as

$$ \mathrm{relu}(\vec{x}) = \max(\alpha\vec{x},\vec{x}), \ 0\leq\alpha<1 $$

This is similar to the ReLU activation we've seen in class, only that it has a small non-zero slope then it's input is negative. Note that it's not strictly differentiable, however it has sub-gradients, defined separately any positive-valued input and for negative-valued input.

TODO: Complete the implementation of the LeakyReLU class in the hw2/layers.py module.

Now using the LeakyReLU, we can trivially define a regular ReLU block as a special case.

TODO: Complete the implementation of the ReLU class in the hw2/layers.py module.

Sigmoid

The sigmoid function $\sigma(x)$ is also sometimes used as an activation function. We have also seen it previously in the context of logistic regression.

The sigmoid function is defined as

$$ \sigma(\vec{x}) = \frac{1}{1+\exp(-\vec{x})}. $$

Hyperbolic Tangent

The hyperbolic tangent function $\tanh(x)$ is a common activation function used when the output should be in the range [-1, 1].

The tanh function is defined as

$$ \tanh(\vec{x}) = \frac{\exp(x)-\exp(-x)}{\exp(x)+\exp(-\vec{x})}. $$

Linear (fully connected) layer

First, we'll implement an affine transform layer, also known as a fully connected layer.

Given an input $\mat{X}$ the layer computes,

$$ \mat{Z} = \mat{X} \mattr{W} + \vec{b} ,~ \mat{X}\in\set{R}^{N\times D_{\mathrm{in}}},~ \mat{W}\in\set{R}^{D_{\mathrm{out}}\times D_{\mathrm{in}}},~ \vec{b}\in\set{R}^{D_{\mathrm{out}}}. $$

Notes:

TODO: Complete the implementation of the Linear class in the hw2/layers.py module.

Cross-Entropy Loss

As you know by know, cross-entropy is a common loss function for classification tasks. In class, we defined it as

$$\ell_{\mathrm{CE}}(\vec{y},\hat{\vec{y}}) = - {\vectr{y}} \log(\hat{\vec{y}})$$

where $\hat{\vec{y}} = \mathrm{softmax}(x)$ is a probability vector (the output of softmax on the class scores $\vec{x}$) and the vector $\vec{y}$ is a 1-hot encoded class label.

However, it's tricky to compute the gradient of softmax, so instead we'll define a version of cross-entropy that produces the exact same output but works directly on the class scores $\vec{x}$.

We can write, $$\begin{align} \ell_{\mathrm{CE}}(\vec{y},\hat{\vec{y}}) &= - {\vectr{y}} \log(\hat{\vec{y}}) = - {\vectr{y}} \log\left(\mathrm{softmax}(\vec{x})\right) \\ &= - {\vectr{y}} \log\left(\frac{e^{\vec{x}}}{\sum_k e^{x_k}}\right) \\ &= - \log\left(\frac{e^{x_y}}{\sum_k e^{x_k}}\right) \\ &= - \left(\log\left(e^{x_y}\right) - \log\left(\sum_k e^{x_k}\right)\right)\\ &= - x_y + \log\left(\sum_k e^{x_k}\right) \end{align}$$

Where the scalar $y$ is the correct class label, so $x_y$ is the correct class score.

Note that this version of cross entropy is also what's provided by PyTorch's nn module.

TODO: Complete the implementation of the CrossEntropyLoss class in the hw2/layers.py module.

Building Models

Now that we have some working Layers, we can build an MLP model of arbitrary depth and compute end-to-end gradients.

First, lets copy an idea from PyTorch and implement our own version of the nn.Sequential Module. This is a Layer which contains other Layers and calls them in sequence. We'll use this to build our MLP model.

TODO: Complete the implementation of the Sequential class in the hw2/layers.py module.

Now, equipped with a Sequential, all we have to do is create an MLP architecture. We'll define our MLP with the following hyperparameters:

So the architecture will be:

FC($D$, $h_1$) $\rightarrow$ ReLU $\rightarrow$ FC($h_1$, $h_2$) $\rightarrow$ ReLU $\rightarrow$ $\cdots$ $\rightarrow$ FC($h_{L-1}$, $h_L$) $\rightarrow$ ReLU $\rightarrow$ FC($h_{L}$, $C$)

We'll also create a sequence of the above MLP and a cross-entropy loss, since it's the gradient of the loss that we need in order to train a model.

TODO: Complete the implementation of the MLP class in the hw2/layers.py module. Ignore the dropout parameter for now.

If the above tests passed then congratulations - you've now implemented an arbitrarily deep model and loss function with end-to-end automatic differentiation!

Questions

TODO Answer the following questions. Write your answers in the appropriate variables in the module hw2/answers.py.

Question 1

Suppose we have a linear (i.e. fully-connected) layer, defined with in_features=1024 and out_features=2048. We apply this layer to an input tensor $\mat{X}$ containing a batch of N=128 samples.

  1. What would then be the shape of the Jacobian tensor of the output of the layer w.r.t. the input $\mat{X}$?

  2. Assuming we're using single-precision floating point (32 bits) to represent our tensors, How many gigabytes of RAM or GPU memory will be required to store the above Jacobian?

$$ \newcommand{\mat}[1]{\boldsymbol {#1}} \newcommand{\mattr}[1]{\boldsymbol {#1}^\top} \newcommand{\matinv}[1]{\boldsymbol {#1}^{-1}} \newcommand{\vec}[1]{\boldsymbol {#1}} \newcommand{\vectr}[1]{\boldsymbol {#1}^\top} \newcommand{\rvar}[1]{\mathrm {#1}} \newcommand{\rvec}[1]{\boldsymbol{\mathrm{#1}}} \newcommand{\diag}{\mathop{\mathrm {diag}}} \newcommand{\set}[1]{\mathbb {#1}} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \newcommand{\pderiv}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\bb}[1]{\boldsymbol{#1}} $$

Part 2: Optimization and Training

In this part we will learn how to implement optimization algorithms for deep networks. Additionally, we'll learn how to write training loops and implement a modular model trainer. We'll use our optimizers and training code to test a few configurations for classifying images with an MLP model.

Implementing Optimization Algorithms

In the context of deep learning, an optimization algorithm is some method of iteratively updating model parameters so that the loss converges toward some local minimum (which we hope will be good enough).

Gradient descent-based methods are by far the most popular algorithms for optimization of neural network parameters. However the high-dimensional loss-surfaces we encounter in deep learning applications are highly non-convex. They may be riddled with local minima, saddle points, large plateaus and a host of very challenging "terrain" for gradient-based optimization. This gave rise to many different methods of performing the parameter updates based on the loss gradients, aiming to tackle these optimization challenges.

The most basic gradient-based update rule can be written as,

$$ \vec{\theta} \leftarrow \vec{\theta} - \eta \nabla_{\vec{\theta}} L(\vec{\theta}; \mathcal{D}) $$

where $\mathcal{D} = \left\{ (\vec{x}^i, \vec{y}^i) \right\}_{i=1}^{M}$ is our training dataset or part of it. Specifically, if we have in total $N$ training samples, then

The intuition behind gradient descent is simple: since the gradient of a multivariate function points to the direction of steepest ascent ("uphill"), we move in the opposite direction. A small step size $\eta$ known as the learning rate is required since the gradient can only serve as a first-order linear approximation of the function's behaviour at $\vec{\theta}$ (recall e.g. the Taylor expansion). However in truth our loss surface generally has nontrivial curvature caused by a high order nonlinear dependency on $\vec{\theta}$. Thus taking a large step in the direction of the gradient is actually just as likely to increase the function value.

The idea behind the stochastic versions is that by constantly changing the samples we compute the loss with, we get a dynamic error surface, i.e. it's different for each set of training samples. This is thought to generally improve the optimization since it may help the optimizer get out of flat regions or sharp local minima since these features may disappear in the loss surface of subsequent batches. The image below illustrates this. The different lines are different 1-dimensional losses for different training set-samples.

Deep learning frameworks generally provide implementations of various gradient-based optimization algorithms. Here we'll implement our own optimization module from scratch, this time keeping a similar API to the PyTorch optim package.

We define a base Optimizer class. An optimizer holds a set of parameter tensors (these are the trainable parameters of some model) and maintains internal state. It may be used as follows:

The exact method of update is implementation-specific for each optimizer and may depend on its internal state. In addition, adding the regularization penalty to the gradient is handled by the optimizer since it only depends on the parameter values (and not the data).

Here's the API of our Optimizer:

Vanilla SGD with Regularization

Let's start by implementing the simplest gradient based optimizer. The update rule will be exacly as stated above, but we'll also add a L2-regularization term to the gradient. Remember that in the loss function, the L2 regularization term is expressed by

$$R(\vec{\theta}) = \frac{1}{2}\lambda||\vec{\theta}||^2_2.$$

TODO: Complete the implementation of the VanillaSGD class in the hw2/optimizers.py module.

Training

Now that we can build a model and loss function, compute their gradients and we have an optimizer, we can finally do some training!

In the spirit of more modular software design, we'll implement a class that will aid us in automating the repetitive training loop code that we usually write over and over again. This will be useful for both training our Layer-based models and also later for training PyTorch nn.Modules.

Here's our Trainer API:

The Trainer class splits the task of training (and evaluating) models into three conceptual levels,

It implements the first two levels. Inheriting classes are expected to implement the single-batch level methods since these are model and/or task specific.

The first thing we should do in order to verify our model, gradient calculations and optimizer implementation is to try to overfit a large model (many parameters) to a small dataset (few images). This will show us that things are working properly.

Let's begin by loading the CIFAR-10 dataset.

Now, let's implement just a small part of our training logic since that's what we need right now.

TODO:

  1. Complete the implementation of the train_batch() method in the LayerTrainer class within the hw2/training.py module.
  2. Update the hyperparameter values in the part2_overfit_hp() function in the hw2/answers.py module. Tweak the hyperparameter values until your model overfits a small number of samples in the code block below. You should get 100% accuracy within a few epochs.

The following code block will use your custom Layer-based MLP implentation, custom Vanilla SGD and custom trainer to overfit the data. The classification accuracy should be 100% within a few epochs.

Now that we know training works, let's try to fit a model to a bit more data for a few epochs, to see how well we're doing. First, we need a function to plot the FitResults object.

TODO:

  1. Complete the implementation of the test_batch() method in the LayerTrainer class within the hw2/training.py module.
  2. Implement the fit() method of the Trainer class within the hw2/training.py module.
  3. Tweak the hyperparameters for this section in the part2_optim_hp() function in the hw2/answers.py module.
  4. Run the following code blocks to train. Try to get above 35-40% test-set accuracy.

Momentum

The simple vanilla SGD update is rarely used in practice since it's very slow to converge relative to other optimization algorithms.

One reason is that naïvely updating in the direction of the current gradient causes it to fluctuate wildly in areas where the loss surface in some dimensions is much steeper than in others. Another reason is that using the same learning rate for all parameters is not a great idea since not all parameters are created equal. For example, parameters associated with rare features should be updated with a larger step than ones associated with commonly-occurring features because they'll get less updates through the gradients.

Therefore more advanced optimizers take into account the previous gradients of a parameter and/or try to use a per-parameter specific learning rate instead of a common one.

Let's now implement a simple and common optimizer: SGD with Momentum. This optimizer takes previous gradients of a parameter into account when updating it's value instead of just the current one. In practice it usually provides faster convergence than the vanilla SGD.

The SGD with Momentum update rule can be stated as follows: $$\begin{align} \vec{v}_{t+1} &= \mu \vec{v}_t - \eta \delta \vec{\theta}_t \\ \vec{\theta}_{t+1} &= \vec{\theta}_t + \vec{v}_{t+1} \end{align}$$

Where $\eta$ is the learning rate, $\vec{\theta}$ is a model parameter, $\delta \vec{\theta}_t=\pderiv{L}{\vec{\theta}}(\vec{\theta}_t)$ is the gradient of the loss w.r.t. to the parameter and $0\leq\mu<1$ is a hyperparameter known as momentum.

Expanding the update rule recursively shows us now the parameter update infact depends on all previous gradient values for that parameter, where the old gradients are exponentially decayed by a factor of $\mu$ at each timestep.

Since we're incorporating previous gradient (update directions), a noisy value of the current gradient will have less effect so that the general direction of previous updates is maintained somewhat. The following figure illustrates this.

TODO:

  1. Complete the implementation of the MomentumSGD class in the hw2/optimizers.py module.
  2. Tweak the learning rate for momentum in part2_optim_hp() the function in the hw2/answers.py module.
  3. Run the following code block to compare to the vanilla SGD.

RMSProp

This is another optmizer that accounts for previous gradients, but this time it uses them to adapt the learning rate per parameter.

RMSProp maintains a decaying moving average of previous squared gradients, $$ \vec{r}_{t+1} = \gamma\vec{r}_{t} + (1-\gamma)\delta\vec{\theta}_t^2 $$ where $0<\gamma<1$ is a decay constant usually set close to $1$, and $\delta\vec{\theta}_t^2$ denotes element-wise squaring.

The update rule for each parameter is then, $$ \vec{\theta}_{t+1} = \vec{\theta}_t - \left( \frac{\eta}{\sqrt{r_{t+1}+\varepsilon}} \right) \delta\vec{\theta}_t $$

where $\varepsilon$ is a small constant to prevent numerical instability. The idea here is to decrease the learning rate for parameters with high gradient values and vice-versa. The decaying moving average prevents accumulating all the past gradients which would cause the effective learning rate to become zero.

TODO:

  1. Complete the implementation of the RMSProp class in the hw2/optimizers.py module.
  2. Tweak the learning rate for RMSProp in part2_optim_hp() the function in the hw2/answers.py module.
  3. Run the following code block to compare to the other optimizers.

Note that you should get better train/test accuracy with Momentum and RMSProp than Vanilla.

Dropout Regularization

Dropout is a useful technique to improve generalization of deep models.

The idea is simple: during the forward pass drop, i.e. set to to zero, the activation of each neuron, with a probability of $p$. For example, if $p=0.4$ this means we drop the activations of 40% of the neurons (on average).

There are a few important things to note about dropout:

  1. It is only performed during training. When testing our model the dropout layers should be a no-op.
  2. In the backward pass, gradients are only propagated back into neurons that weren't dropped during the forward pass.
  3. During testing, the activations must be scaled since the expected value of each neuron during the training phase is now $1-p$ times it's original expectation. Thus, we need to scale the test-time activations by $1-p$ to match. Equivalently, we can scale the train time activations by $1/(1-p)$.

TODO:

  1. Complete the implementation of the Dropout class in the hw2/layers.py module.
  2. Finish the implementation of the MLP's __init__() method in the hw2/layers.py module. If dropout>0 you should add a Dropout layer after each ReLU.

To see whether dropout really improves generalization, let's take a small training set (small enough to overfit) and a large test set and check whether we get less overfitting and perhaps improved test-set accuracy when using dropout.

TODO: Tweak the hyperparameters for this section in the part2_dropout_hp() function in the hw2/answers.py module. Try to set them so that the first model (with dropout=0) overfits. You can disable the other dropout options until you tune the hyperparameters. We can then see the effect of dropout for generalization.

Questions

TODO Answer the following questions. Write your answers in the appropriate variables in the module hw2/answers.py.

Question 1

Regarding the graphs you got for the three dropout configurations:

  1. Explain the graphs of no-dropout vs dropout. Do they match what you expected to see?

    • If yes, explain why and provide examples based on the graphs.
    • If no, explain what you think the problem is and what should be modified to fix it.

  2. Compare the low-dropout setting to the high-dropout setting and explain based on your graphs.

Question 2

When training a model with the cross-entropy loss function, is it possible for the test loss to increase for a few epochs while the test accuracy also increases?

If it's possible explain how, if it's not explain why not.

$$ \newcommand{\mat}[1]{\boldsymbol {#1}} \newcommand{\mattr}[1]{\boldsymbol {#1}^\top} \newcommand{\matinv}[1]{\boldsymbol {#1}^{-1}} \newcommand{\vec}[1]{\boldsymbol {#1}} \newcommand{\vectr}[1]{\boldsymbol {#1}^\top} \newcommand{\rvar}[1]{\mathrm {#1}} \newcommand{\rvec}[1]{\boldsymbol{\mathrm{#1}}} \newcommand{\diag}{\mathop{\mathrm {diag}}} \newcommand{\set}[1]{\mathbb {#1}} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \newcommand{\pderiv}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\bb}[1]{\boldsymbol{#1}} $$

Part 3: Convolutional Architectures

In this part we will explore convolution networks and the effects of their architecture on accuracy. We'll implement a common block-based deep CNN pattern and we'll perform various experiments on it while varying the architecture. Then we'll implement our own custom architecture to see whether we can get high classification results on a large subset of CIFAR-10.

Training will be performed on GPU.

Convolutional layers and networks

Convolutional layers are the most essential building blocks of the state of the art deep learning image classification models and also play an important role in many other tasks. As we saw in the tutorial, when applied to images, convolutional layers operate on and produce volumes (3D tensors) of activations.

A convenient way to interpret convolutional layers for images is as a collection of 3D learnable filters, each of which operates on a small spatial region of the input volume. Each filter is convolved with the input volume ("slides over it"), and a dot product is computed at each location followed by a non-linearity which produces one activation. All these activations produce a 2D plane known as a feature map. Multiple feature maps (one for each filter) comprise the output volume.

A crucial property of convolutional layers is their translation equivariance, i.e. shifting the input results in and equivalently shifted output. This produces the ability to detect features regardless of their spatial location in the input.

Convolutional network architectures usually follow a pattern basic repeating blocks: one or more convolution layers, each followed by a non-linearity (generally ReLU) and then a pooling layer to reduce spatial dimensions. Usually, the number of convolutional filters increases the deeper they are in the network. These layers are meant to extract features from the input. Then, one or more fully-connected layers is used to combine the extracted features into the required number of output class scores.

Building convolutional networks with PyTorch

PyTorch provides all the basic building blocks needed for creating a convolutional arcitecture within the torch.nn package. Let's use them to create a basic convolutional network with the following architecture pattern:

[(CONV -> ACT)*P -> POOL]*(N/P) -> (FC -> ACT)*M -> FC

Here $N$ is the total number of convolutional layers, $P$ specifies how many convolutions to perform before each pooling layer and $M$ specifies the number of hidden fully-connected layers before the final output layer.

TODO: Complete the implementaion of the ConvClassifier class in the hw2/cnn.py module. Use PyTorch's nn.Conv2d and nn.MaxPool2d for the convolution and pooling layers. It's recommended to implement the missing functionality in the order of the class' methods.

Let's load CIFAR-10 again to use as our dataset.

Now as usual, as a sanity test let's make sure we can overfit a tiny dataset with our model. But first we need to adapt our Trainer for PyTorch models.

TODO: Complete the implementaion of the TorchTrainer class in the hw2/training.py module.

Residual Networks

A very common addition to the basic convolutional architecture described above are shortcut connections. First proposed by He et al. (2016), this simple addition has been shown to be crucial ingredient in order to achieve effective learning with very deep networks. Virtually all state of the art image classification models from recent years use this technique.

The idea is to add an shortcut, or skip, around every two or more convolutional layers:

This adds an easy way for the network to learn identity mappings: set the weight values to be very small. The consequence is that the convolutional layers to learn a residual mapping, i.e. some delta that is applied to the identity map, instead of actually learning a completely new mapping from scratch.

Lets start by implementing a general residual block, representing a structure similar to the above diagrams. Our residual block will be composed of:

TODO: Complete the implementation of the ResidualBlock's __init__() method in the hw2/cnn.py module.

Now, based on the ResidualBlock, we'll implement our own variation of a residual network (ResNet), with the following architecture:

[-> (CONV -> ACT)*P -> POOL]*(N/P) -> (FC -> ACT)*M -> FC
 \------- SKIP ------/

Note that $N$, $P$ and $M$ are as before, however now $P$ also controls the number of convolutional layers to add a skip-connection to.

Bottleneck Blocks

In the ResNet Block diagram shown above, the right block is called a bottleneck block. This type of block is mainly used deep in the network, where the feature space becomes increasingly high-dimensional (i.e. there are many channels).

Instead of applying a KxK conv layer on the original input channels, a bottleneck block first projects to a lower number of features (channels), applies the KxK conv on the result, and then projects back to the original feature space. Both projections are performed with 1x1 convolutions.

TODO: Complete the implementation of the ResidualBottleneckBlock in the hw2/cnn.py module.

TODO: Complete the implementation of the ResNetClassifier class in the hw2/cnn.py module. You must use your ResidualBlocks to group together every $P$ convolutional layers.

Experimenting with model architectures

You will now perform a series of experiments that train various model configurations on a much larger part of the CIFAR-10 dataset.

To perform the experiments, you'll need to use a machine with a GPU since training time might be too long otherwise.

Note about running on GPUs

Here's an example of running a forward pass on the GPU (assuming you're running this notebook on a GPU-enabled machine).

Notice how we called .to(device) on both the model and the input tensor. Here the device is a torch.device object that we created above. If an nvidia GPU is available on the machine you're running this on, the device will be 'cuda'. When you run .to(device) on a model, it recursively goes over all the model parameter tensors and copies their memory to the GPU. Similarly, calling .to(device) on the input image also copies it.

In order to train on a GPU, you need to make sure to move all your tensors to it. You'll get errors if you try to mix CPU and GPU tensors in a computation.

Notes on using course servers

First, please read the course servers guide carefully.

To run the experiments on the course servers, you can use the py-sbatch.sh script directly to perform a single experiment run in batch mode (since it runs python once), or use the srun command to do a single run in interactive mode. For example, running a single run of experiment 1 interactively (after conda activate of course):

srun -c 2 --gres=gpu:1 --pty python -m hw2.experiments run-exp -n test -K 32 64 -L 2 -P 2 -H 100

To perform multiple runs in batch mode with sbatch (e.g. for running all the configurations of an experiments), you can create your own script based on py-sbatch.sh and invoke whatever commands you need within it.

Please don't request more than 2 CPU cores and 1 GPU device for your runs. The code won't be able to utilize more than that anyway, so you'll see no performance gain if you do. It will only cause delays for other students using the servers.

General notes for running experiments

Experiment 1: Network depth and number of filters

In this part we will test some different architecture configurations based on our ConvClassifier and ResNetClassifier. Specifically, we want to try different depths and number of features to see the effects these parameters have on the model's performance.

To do this, we'll define two extra hyperparameters for our model, K (filters_per_layer) and L (layers_per_block).

For example, if K=[32, 64] and L=2 it means we want two conv layers with 32 filters followed by two conv layers with 64 filters. If we also use pool_every=3, the feature-extraction part of our model will be:

Conv(X,32)->ReLu->Conv(32,32)->ReLU->Conv(32,64)->ReLU->MaxPool->Conv(64,64)->ReLU

We'll try various values of the K and L parameters in combination and see how each architecture trains. All other hyperparameters are up to you, including the choice of the optimization algorithm, the learning rate, regularization and architecture hyperparams such as pool_every and hidden_dims. Note that you should select the pool_every parameter wisely per experiment so that you don't end up with zero-width feature maps.

You can try some short manual runs to determine some good values for the hyperparameters or implement cross-validation to do it. However, the dataset size you test on should be large. Use at least ~20000 training images and ~6000 validation images.

The important thing is that you state what you used, how you decided on it, and explain your results based on that.

First we need to write some code to run the experiment.

TODO:

  1. Implement the run_experiment() function in the hw2/experiments.py module.
  2. If you haven't done so already, it would be an excellent idea to implement the early stopping feature of the Trainer class.

The following block tests that your implementation works. It's also meant to show you that each experiment run creates a result file containing the parameters to reproduce and the FitResult object for plotting.

We'll use the following function to load multiple experiment results and plot them together.

Experiment 1.1: Varying the network depth (L)

First, we'll test the effect of the network depth on training.

Configuratons:

So 8 different runs in total.

Naming runs: Each run should be named exp1_1_L{}_K{} where the braces are placeholders for the values. For example, the first run should be named exp1_1_L2_K32.

TODO: Run the experiment on the above configuration with the ConvClassifier model. Make sure the result file names are as expected. Use the following blocks to display the results.

Experiment 1.2: Varying the number of filters per layer (K)

Now we'll test the effect of the number of convolutional filters in each layer.

Configuratons:

So 12 different runs in total. To clarify, each run K takes the value of a list with a single element.

Naming runs: Each run should be named exp1_2_L{}_K{} where the braces are placeholders for the values. For example, the first run should be named exp1_2_L2_K32.

TODO: Run the experiment on the above configuration with the ConvClassifier model. Make sure the result file names are as expected. Use the following blocks to display the results.

Experiment 1.3: Varying both the number of filters (K) and network depth (L)

Now we'll test the effect of the number of convolutional filters in each layer.

Configuratons:

So 4 different runs in total. To clarify, each run K takes the value of an array with a three elements.

Naming runs: Each run should be named exp1_3_L{}_K{}-{}-{} where the braces are placeholders for the values. For example, the first run should be named exp1_3_L1_K64-128-256.

TODO: Run the experiment on the above configuration with the ConvClassifier model. Make sure the result file names are as expected. Use the following blocks to display the results.

Experiment 1.4: Adding depth with Residual Networks

Now we'll test the effect of skip connections on the training and performance.

Configuratons:

So 6 different runs in total.

Naming runs: Each run should be named exp1_4_L{}_K{}-{}-{} where the braces are placeholders for the values.

TODO: Run the experiment on the above configuration with the ResNetClassifier model. Make sure the result file names are as expected. Use the following blocks to display the results.

Experiment 2: Custom network architecture

In this part you will create your own custom network architecture based on the ConvClassifier you've implemented.

Try to overcome some of the limitations your experiment 1 results, using what you learned in the course.

You are free to add whatever you like to the model, for instance

Just make sure to keep the model's init API identical (or maybe just add parameters).

TODO: Implement your custom architecture in the YourCodeNet class within the hw2/cnn.py module.

Experiment 2 Configuration

Run your custom model on at least the following:

Configuratons:

So 4 different runs in total. To clarify, each run K takes the value of an array with a three elements.

If you want, you can add some extra runs following the same pattern. Try to see how deep a model you can train.

Naming runs: Each run should be named exp2_L{}_K{}-{}-{}-{} where the braces are placeholders for the values. For example, the first run should be named exp2_L3_K32-64-128.

TODO: Run the experiment on the above configuration with the YourCodeNet model. Make sure the result file names are as expected. Use the following blocks to display the results.

Questions

TODO Answer the following questions. Write your answers in the appropriate variables in the module hw2/answers.py.

Question 1

Consider the bottleneck block from the right side of the ResNet diagram above. Compare it to a regular block that performs a two 3x3 convs directly on the 256-channel input (i.e. as shown in the left side of the diagram, with a different number of channels). Explain the differences between the regular block and the bottleneck block in terms of:

  1. Number of parameters. Calculate the exact numbers for these two examples.
  2. Number of floating point operations required to compute an output (qualitative assessment).
  3. Ability to combine the input: (1) spatially (within feature maps); (2) across feature maps.

Question 2

Analyze your results from experiment 1.1. In particular,

  1. Explain the effect of depth on the accuracy. What depth produces the best results and why do you think that's the case?
  2. Were there values of L for which the network wasn't trainable? what causes this? Suggest two things which may be done to resolve it at least partially.

Question 3

Analyze your results from experiment 1.2. In particular, compare to the results of experiment 1.1.

Question 4

Analyze your results from experiment 1.3.

Question 5

Analyze your results from experiment 1.4. Compare to experiment 1.1 and 1.3.

Question 6

  1. Explain your modifications to the architecture which you implemented in the YourCodeNet class.
  2. Analyze the results of experiment 2. Compare to experiment 1.